20 ANALYTICAL MECHANICS 3 Find the horizontal force which will keep in equilibrium a weight of 150 pounds on a smooth inclined plane which makes CO with the horizon. 4. A ring of weight TF is suspended by means of a string of length J, the ends of which are attached to two points on the same horizontal line. Find the tensile force of the string if the distance between its ends is d. Also discuss the limiting cases in which I approaches d or becomes very large compared with it. 5. A body of weight W is suspended by two strings of lengths Zi and I* The upper end of each string is attached to a fixed point in the same horizontal line. Find the tensile forces in the strings if the distance between the two points is d. 6. A weight is suspended by four equal strings, the upper ends of which are attached to the vertices of a horizontal square. Find the tensile forces in the strings. 7. A particle is in equilibrium on a smooth inclined plane under the action of two equal forces, the one acting along the plane up wards and the other horizontally. Find the inclination of the plane. 8. Apply the conditions of equilibrium to find the magnitude and direction of the resultant of a number of forces acting upon a particle. 9. Two spheres of equal radius and equal weight are in equilibrium in a smooth hemispherical bowl; find the reactions between the two spheres and between the spheres and the bowl. 10. The ends of a string, 60 cm. long, are fastened to two points in the same horizontal line and at a distance of 40 cm. apart; two weights are hung from points in the string 25 cm. and 20 cm. from the ends. Find the ratio of the weights if the part of the string between them is horizontal. 11. A single triangular truss of 24 feet span and 5 feet depth supports a load of 3 tons at the apex. Find the forces acting on the rafters and the tie rod. 12. A particle of weight W can be kept in equilibrium upon a smooth inclined plane by a force FI acting horizontally; it can also be kept in equilibrium by a force F2 acting parallel to the plane. Express W in terms of F\ and F$. 13. In the Mowing arrangements of pulleys find the relation between F and W.